What is the fractional Laplacian? A comparative review with new results

Lischke, Anna; Pang, Guofei; Gulian, Mamikon; Song, Fangying; Glusa, Christian; Zheng, Xiaolai; Mao, Zhiping; Cai, Wei; Meerschaert, Mark M.; Ainsworth, Mark; Karniadakis, George Em

doi:10.1016/j.jcp.2019.109009

Cited by 332 publications

(200 citation statements)

References 113 publications

Supporting

Mentioning

179

Contrasting

Unclassified

Order By: Relevance

“…where ∂ 2 ∂x 2 + 1 x ∂ ∂x α 2 is a fractional Laplacian, defined by its Fourier transform [Kwaśnicki, 2017, Lischke et al, 2020. Note that the rotational symmetry of the problem allows us to write the Laplacian in terms of just the radial coordinate x, and ignore the angular coordinate.…”

Section: Analytical Model In Two Dimensionsmentioning

confidence: 99%

Isolation by Distance in Populations with Power-law Dispersal

2020

Preprint

View full text Add to dashboard Cite

Limited dispersal results in isolation by distance in spatially structured populations, in which individuals found further apart tend to be less related to each other. Models of populations undergoing short-range dispersal predict a close relation between the distance individuals disperse and the length scale over which two sampled individuals are likely to be closely related. In this work, we study the effect of long jumps on patterns of isolation by distance by replacing the typical short-range dispersal kernel with a long-range, power-law kernel. We find that incorporating long jumps leads to a slower decay of relatedness with distance, and that the quantitative form of this slow decay contains visible signatures of the underlying dispersal process. K 1 (y|t) = 1 2π ∞ −∞ dk exp (−iky − D α t|k| α ) .(1)

show abstract

Section: Analytical Model In Two Dimensionsmentioning

confidence: 99%

Isolation by Distance in Populations with Power-law Dispersal

2020

Preprint

View full text Add to dashboard Cite

show abstract

“…In a similar vein to the fractional derivatives, a multitude of fractional Laplacian (FL) definitions have been proposed over the past few decades but, a consensus on the most appropriate definition for an application is yet to be reached [17].…”

Section: B Fractional Laplacian (Fl)mentioning

confidence: 99%

“…Of special interest to our work are two variants: the spectral fractional Laplacian [17] which is defined as follows:…”

Section: B Fractional Laplacian (Fl)mentioning

confidence: 99%

See 1 more Smart Citation

Discrete Laplacian Operator and Its Applications in Signal Processing

2020

View full text Add to dashboard Cite

Fractional calculus has increased in popularity in recent years, as the number of its applications in different fields has increased. Compared to the traditional operations in calculus (integration and differentiation) which are uniquely defined, the fractional-order operators have numerous definitions. Furthermore, a consensus on the most suitable definition for a given task is yet to be reached. Fractional operators are defined as continuous operators and their implementation requires a discretization step. In this article, we propose a discrete fractional Laplacian as a matrix operator. The proposed operator is real (non-complex) which makes it computationally efficient. The construction of the proposed fractional Laplacian utilizes the DCT transform avoiding the complexity associated with the discretization step which is typical in the constructions based on signal processing. We demonstrate the utility of the proposed operator on a number of data modeling and image processing tasks. INDEX TERMS Fractional-Laplacian, discrete operator, image-processing, trend-filtering, fractional calculus.

show abstract

“…However, in bounded domains, boundary conditions must be incorporated in these characterizations in mathematically distinct ways, and there is currently no consensus in the literature as to which definition of the fractional Laplacian in bounded domains is most appropriate for a given application. In a recent article, Lischke et al compared the Poisson equation with different definitions of the fractional Laplacian by numerical studies. The authors presented that the Riesz integral definition admits a nonlocal boundary condition, where the value of a function must be prescribed on the entire exterior of the domain in order to compute its fractional Laplacian.…”

Section: Introductionmentioning

confidence: 99%

The finite element method for fractional diffusion with spectral fractional Laplacian

Cheng

2020

Math Methods in App Sciences

View full text Add to dashboard Cite

This paper focuses on the finite element method for Caputo‐type parabolic equation with spectral fractional Laplacian, where the time derivative is in the sense of Caputo with order in (0,1) and the spatial derivative is the spectral fractional Laplacian. The time discretization is based on the Hadamard finite‐part integral (or the finite‐part integral in the sense of Hadamard), where the piecewise linear interpolation polynomials are used. The spatial fractional Laplacian is lifted to the local spacial derivative by using the Caffarelli–Silvestre extension, where the finite element method is used. Full‐discretization scheme is constructed. The convergence and error estimates are obtained. Finally, numerical experiments are presented which support the theoretical results.

show abstract

What is the fractional Laplacian? A comparative review with new results

Cited by 332 publications

References 113 publications

Isolation by Distance in Populations with Power-law Dispersal

Isolation by Distance in Populations with Power-law Dispersal

Discrete Laplacian Operator and Its Applications in Signal Processing

The finite element method for fractional diffusion with spectral fractional Laplacian

Contact Info

Product

Resources

About